Is it true that Omar ibn Al-Khattab prayed to Allah to die as a martyr and in the Prophet's Mosque? And is it permissible for a person supplicating to ask Allah to marry him to a specific girl in this world or in the Hereafter?

Umar's supplication - may Allah be pleased with him - to be granted martyrdom in Allah's cause and to die in the city of His Prophet, is a legitimate and permissible supplication. This differs from supplicating for a married woman to become the supplicant's wife, as this is an transgression and impermissible, for it entails the separation of spouses, the corruption of relationships, and the ruin of children.

A Muslim does not know where the good lies, and he must submit his affair to Allah the Almighty, entrusting Him with His decree and destiny. He should also seek Allah's guidance (istikharah) in important matters; it was narrated that the Prophet (peace be upon him) used to teach his companions istikharah as he taught them a chapter from the Quran, saying: "If any one of you intends to do a matter, let him pray two rak'ahs other than the obligatory prayers and then say: 'O Allah, I seek Your guidance by Your knowledge, and I seek ability by Your power, and I ask You from Your immense bounty. For You are able and I am not, You know and I do not, and You are the Knower of the unseen. O Allah, if You know that this matter is good for me in my religion, my livelihood, and my ultimate outcome - or he said: in my present and future life - then decree it for me and make it easy for me and then bless me in it. And if You know that this matter is bad for me in my religion, my livelihood, and my ultimate outcome - or he said: in my present and future life - then turn it away from me and turn me away from it, and decree for me good wherever it may be, and then make me content with it.' He (the narrator) said: 'Then he should name his need.'"